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{{DISPLAYTITLE:General m<sub>n</sub>-type image filters}}
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{{Linear analog electronic filter|filter1=hide|filter3=hide}}


These filters are electrical wave filters designed using the [[image impedance|image method]]. They are an invention of [[Otto Zobel]] at [[AT&T Corp.]].<ref>Zobel, 1923</ref> They are a generalisation of the [[m-type filter]] in that a transform is applied that modifies the transfer function while keeping the image impedance unchanged.  For filters that have only one [[stopband]] there is no distinction with the m-type filter. However, for a filter that has multiple stopbands, there is the possibility that the form of the transfer function in each stopband can be different. For instance, it may be required to filter one band with the sharpest possible cut-off, but in another to minimise phase distortion while still achieving some attenuation. If the form is identical at each transition from passband to stopband the filter will be the same as an  m-type filter ([[k-type filter]] in the limiting case of ''m''=1).  If they are different, then the general case described here pertains.
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The k-type filter acts as a [[prototype filter|prototype]] for producing the general ''m''<sub>n</sub> designs. For any given desired bandform there are two classes of m<sub>n</sub> transformation that can be applied, namely, the mid-series and mid-shunt derived sections; this terminology being more fully explained in the [[m-derived filter]] article. Another feature of m-type filters that also applies in the general case is that a half section will have the original k-type image impedance on one side only. The other port will present a new image impedance. The two transformations have equivalent transfer functions but different image impedances and circuit topology.
{|
|-valign="top"
|__TOC__
|[[File:Image filter bandform terminology.svg|thumb|right|400px|Bandform diagram showing frequency response of a general image filter.  The ''ω''<sub>c</sub> are the critical frequencies (the frequency where cut-off begins) and the ''ω''<sub>∞</sub> are the poles of attenuation in the stop bands.]]
|}
{{complex Z}}
==Mid-series multiple stopband==
[[File:Image filter sections 1.svg|right|200px]]
If ''Z'' and ''Y'' are the series impedance and shunt admittance of a constant ''k'' half section and;
 
:<math>Z = Z_1 + Z_2 + Z_3 + \dots + Z_N</math>
 
:where ''Z''<sub>1</sub>, Z<sub>2</sub> etc are a cascade of antiresonators,
 
the transformed series impedance for a mid-series derived filter becomes;
 
:<math>Z_{m_n} = m_1Z_1 + m_2Z_2 + m_3Z_3 + \dots + m_NZ_N</math>
 
Where the ''m<sub>n</sub>'' are arbitrary positive coefficients. For an invariant image impedance ''Z''<sub>iT</sub> and invariant bandform (that is, invariant cut-off frequencies ''ω''<sub>''c''</sub>) the transformed shunt admittance, expressed in terms of ''Z<sub>m<sub>n</sub></sub>'', is given by;
 
:<math>Y_{m_n} = \frac {Z_{m_n}} {k^2 + Z^2 - Z_{m_n}^2}</math>
 
:where <math>k = \sqrt{\frac{Z}{Y}} </math> and is a constant by definition. When the ''m<sub>n</sub>'' are all equal this reduces to the expression for an m-type filter and where they are all equal to one it reduces further to the ''k''-type filter.
[[File:Series derived multi-stop mn type filter.svg|left|thumb|350px|2-bandstop mid-series derived m<sub>1</sub>, m<sub>2</sub> filter half-section]]
A result of this relationship is that the ''N'' antiresonators in ''Z<sub>m<sub>n</sub></sub>'' will transform into 2''N'' resonators in ''Y<sub>m<sub>n</sub></sub>''. The coefficients ''m<sub>n</sub>'' can be adjusted by the designer to set the frequency of one of the two poles of attenuation, ω<sub>∞</sub>, in each stopband. The second pole of attenuation is dependant and cannot be set separately.
 
===Special cases===
 
In the case of a filter with a stopband extending to zero frequency, one of the antiresonators in ''Z'' will reduce to a single inductor. In this case the resonators in ''Y<sub>m<sub>n</sub></sub>'' are reduced by one to 2''N''-1.  Similarly, for a filter with a stopband extending to infinity, one antiresonator will reduce to a single capacitor and the resonators will again be reduced by one.  In a filter where both conditions pertain, the number of resonators will be 2''N''-2. For these end stopbands, there is only one pole of attenuation in each, as would be expected from the reduced number of resonators. These forms are the maximum allowable complexity while maintaining invariance of bandform and one image impedance.
{{clear}}
 
==Mid-shunt multiple stopband==
[[File:Shunt derived multi-stop mn type filter.svg|thumb|400px|2-bandstop mid-shunt derived m1, m2 filter half-section. Frequency response is equivalent to the corresponding mid-series derived filter]]
By [[dual impedance|dual]] analogy, the shunt derived filter starts from;
 
:<math>Y_{m_n} = m_1Y_1 + m_2Y_2 + m_3Y_3 + \dots + m_NY_N</math>
 
For an invariant image admittance ''Y''<sub>iΠ</sub> and invariant bandform the transformed series impedance is given by;
 
:<math>Z_{m_n} = \frac {Y_{m_n}} {k^2 + Y^2 - Y_{m_n}^2}</math>
{{clear}}
 
==Simple bandpass section==
[[File:General image bandpass filter.svg|left|thumb|450px|General image bandpass filter, mid-shunt derived]]
The bandpass filter can be characterised as a 2-bandstop filter with ''ω''<sub>c</sub>&nbsp;=&nbsp;0 for the lower critical frequency of the lower band  and ''ω''<sub>c</sub>&nbsp;=&nbsp;∞ for the upper critical frequency of the upper band.  The two resonators reduce to an inductor and a capacitor respectively. The number of antiresonators reduces to two.
[[File:Image bandpass filter with one pole af attenuation.svg|right|thumb|375px|Image bandpass filter with ''ω''<sub>∞1</sub> set to zero and ''ω''<sub>∞2</sub> set to correspond to ''ω'<span/>''<sub>c1</sub>.]]
<br style="clear:left;"/>
If, however, ''ω''<sub>∞1</sub> is set to zero (that is, there is no pole of attenuation in the lower stopband) and ''ω''<sub>∞2</sub> is set to correspond to the upper critical frequency ''ω'<span/>''<sub>c1</sub>, then a particularly simple form of the bandpass filter is obtained consisting of just antiresonators coupled by capacitors. This was a popular topology for multi-section band-pass filters due its low component count, particularly of inductors.<ref>Matthaei, p425</ref><ref>Bray, p63</ref>  Many other such reduced forms are possible by setting one of the poles of attenuation to correspond to one of the critical frequencies for various classes of basic filter.<ref>Zobel, pp42-43</ref>
{{clear}}
 
==See also==
[[File:Bandform template.svg|thumb|left|600px|'''Filter bandforms''': see, [[low-pass]], [[high-pass]], [[band-pass]], [[band-stop]].]]
{{clear}}
*[[Image impedance]]
*[[Constant k filter]]
*[[m-derived filter]]
*[[mm'-type filter]]
*[[Composite image filter]]
 
==Notes==
{{reflist}}
 
==References==
<div class="references-small">
:* Zobel, O. J.,''Theory and Design of Uniform and Composite Electric Wave Filters'', Bell Systems Technical Journal, '''Vol. 2''' (1923), pp. 1-46.
:* Mathaei, Young, Jones ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'' McGraw-Hill 1964.
:* Bray, J, ''Innovation and the Communications Revolution'', Institution of Electrical Engineers, 2002 ISBN 0-85296-218-5
</div>
 
[[Category:Image impedance filters]]
[[Category:Electronic filter topology]]

Latest revision as of 10:13, 11 November 2014

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